Riemann surfaces with maximal real symmetry

Let S   be a compact Riemann surface of genus g>1g>1, and let τ:S→Sτ:S→S be any anti-conformal automorphism of S, of order 2. Such an anti-conformal involution is known as a symmetry of S, and the species of all conjugacy classes of all symmetries of S constitute what is known as the symmetry type of S. The surface S is said to have maximal real symmetry   if it admits a symmetry τ:S→Sτ:S→S such that the compact Klein surface S/τS/τ has maximal symmetry (which means that S/τS/τ has the largest possible number of automorphisms with respect to its genus). If τ   has fixed points, which is the only case we consider here, then the maximum number of automorphisms of S/τS/τ is 12(g−1)12(g−1). In the first part of this paper, we develop a computational procedure to compute the symmetry type of every Riemann surface of genus g   with maximal real symmetry, for given small values of g>1g>1. We have used this to find all of them for 1